Some combinatorial identities appearing in the calculation of the cohomology of Siegel modular varieties

TL;DR

This paper introduces a geometric and combinatorial approach to complex identities involving discrete series characters, which are crucial in calculating the cohomology of Siegel modular varieties, without relying on advanced representation theory.

Contribution

It provides a purely combinatorial and geometric method to understand identities in the cohomology of Siegel modular varieties, simplifying previous complex algebraic approaches.

Findings

Uses Coxeter complex of the symmetric group for combinatorial analysis

Derives identities relevant to intersection cohomology calculations

Offers a new perspective avoiding deep representation theory

Abstract

In the computation of the intersection cohomology of Shimura varieties, or of the $L^2$ cohomology of equal rank locally symmetric spaces, combinatorial identities involving averaged discrete series characters of real reductive groups play a large technical role. These identities can become very complicated and are not always well-understood (see for example the appendix of [8]). We propose a geometric approach to these identities in the case of Siegel modular varieties using the combinatorial properties of the Coxeter complex of the symmetric group. Apart from some introductory remarks about the origin of the identities, our paper is entirely combinatorial and does not require any knowledge of Shimura varieties or of representation theory.